Morphological skeleton

In , morphological skeleton is a skeleton (or medial axis) representation of a shape or , computed by means of morphological operators. Morphological skeletons are of two kinds: Those defined by means of morphological openings, from which the original shape can be reconstructed, Those computed by means of the hit-or-miss transform, which preserve the shape's topology. In (Lantuéjoul 1977), Lantuéjoul derived the following morphological formula for the skeleton of a continuous binary image : where and are the morphological erosion and opening, respectively, is an open ball of radius , and is the closure of . Let , , be a family of shapes, where B is a structuring element, and where o denotes the origin. The variable n is called the size of the structuring element. Lantuéjoul's formula has been discretized as follows. For a discrete binary image , the skeleton S(X) is the union of the skeleton subsets , , where: The original shape X can be reconstructed from the set of skeleton subsets as follows: Partial reconstructions can also be performed, leading to opened versions of the original shape: Let be the translated version of to the point z, that is, . A shape centered at z is called a maximal disk in a set A when: and if, for some integer m and some point y, , then . Each skeleton subset consists of the centers of all maximal disks of size n. Morphological Skeletonization can be considered as a controlled erosion process. This involves shrinking the image until the area of interest is 1 pixel wide. This can allow quick and accurate image processing on an otherwise large and memory intensive operation. A great example of using skeletonization on an image is processing fingerprints. This can be quickly accomplished using bwmorph; a built-in Matlab function which will implement the Skeletonization Morphology technique to the image. The image to the right shows the extent of what skeleton morphology can accomplish. Given a partial image, it is possible to extract a much fuller picture.